Mladen Bestvina (born 1959) [1] is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.
Mladen Bestvina (born 1959) [1] is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.
Mladen Bestvina is a three-time medalist at the International Mathematical Olympiad (two silver medals in 1976 and 1978 and a bronze medal in 1977). [2] He received a B. Sc. in 1982 from the University of Zagreb. [3] He obtained a PhD in Mathematics in 1984 at the University of Tennessee under the direction of John Walsh. [4] He was a visiting scholar at the Institute for Advanced Study in 1987-88 and again in 1990–91. [5] Bestvina had been a faculty member at UCLA, and joined the faculty in the Department of Mathematics at the University of Utah in 1993. [6] He was appointed a Distinguished Professor at the University of Utah in 2008. [6] Bestvina received the Alfred P. Sloan Fellowship in 1988–89 [7] [8] and a Presidential Young Investigator Award in 1988–91. [9]
Bestvina gave an invited address at the International Congress of Mathematicians in Beijing in 2002, [10] , was topology section member in Madrid in 2006 [11] and gave a plenary lecture at virtual ICM 2022. [12] He also gave a Unni Namboodiri Lecture in Geometry and Topology at the University of Chicago. [13]
Bestvina served as an Editorial Board member for the Transactions of the American Mathematical Society [14] and as an associate editor of the Annals of Mathematics. [15] Currently he is an editorial board member for Duke Mathematical Journal, [16] Geometric and Functional Analysis, [17] Geometry and Topology, [18] the Journal of Topology and Analysis, [19] Groups, Geometry and Dynamics, [20] Michigan Mathematical Journal, [21] Rocky Mountain Journal of Mathematics, [22] and Glasnik Matematicki. [23]
In 2012 he became a fellow of the American Mathematical Society. [24] Since 2012, he has been a correspondent member of the HAZU (Croatian Academy of Science and Art). [1]
A 1988 monograph of Bestvina [25] gave an abstract topological characterization of universal Menger compacta in all dimensions; previously only the cases of dimension 0 and 1 were well understood. John Walsh wrote in a review of Bestvina's monograph: 'This work, which formed the author's Ph.D. thesis at the University of Tennessee, represents a monumental step forward, having moved the status of the topological structure of higher-dimensional Menger compacta from one of "close to total ignorance" to one of "complete understanding".' [26]
In a 1992 paper Bestvina and Feighn obtained a Combination Theorem for word-hyperbolic groups. [27] The theorem provides a set of sufficient conditions for amalgamated free products and HNN extensions of word-hyperbolic groups to again be word-hyperbolic. The Bestvina–Feighn Combination Theorem became a standard tool in geometric group theory and has had many applications and generalizations (e.g. [28] [29] [30] [31] ).
Bestvina and Feighn also gave the first published treatment of Rips' theory of stable group actions on R-trees (the Rips machine) [32] In particular their paper gives a proof of the Morgan–Shalen conjecture [33] that a finitely generated group G admits a free isometric action on an R-tree if and only if G is a free product of surface groups, free groups and free abelian groups.
A 1992 paper of Bestvina and Handel introduced the notion of a train track map for representing elements of Out(Fn). [34] In the same paper they introduced the notion of a relative train track and applied train track methods to solve [34] the Scott conjecture, which says that for every automorphism α of a finitely generated free group Fn the fixed subgroup of α is free of rank at most n. Since then train tracks became a standard tool in the study of algebraic, geometric and dynamical properties of automorphisms of free groups and of subgroups of Out(Fn). Examples of applications of train tracks include: a theorem of Brinkmann [35] proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and Groves [36] that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups; [37] and others.
Bestvina, Feighn and Handel later proved that the group Out(Fn) satisfies the Tits alternative, [38] [39] settling a long-standing open problem.
In a 1997 paper [40] Bestvina and Brady developed a version of discrete Morse theory for cubical complexes and applied it to study homological finiteness properties of subgroups of right-angled Artin groups. In particular, they constructed an example of a group which provides a counter-example to either the Whitehead asphericity conjecture or to the Eilenberg−Ganea conjecture, thus showing that at least one of these conjectures must be false. Brady subsequently used their Morse theory technique to construct the first example of a finitely presented subgroup of a word-hyperbolic group that is not itself word-hyperbolic. [41]
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